In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 19 Jul 2010 01:13 herbzet <herbzet(a)gmail.com> writes: > Nah -- implication is truth-preserving, by definition. A system may well be consistent even if some of its axioms are false. We are here presupposing the formal language the system is formulated in has some intended interpretation so that it makes sense to speak of truth or falsity. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 19 Jul 2010 01:20 herbzet <herbzet(a)gmail.com> writes: > If the system is consistent, then there is a structure in which > B -> A is true. Sure, but what of it? Whenever we speak of truth or falsity we must have some specific interpretation in mind. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 19 Jul 2010 01:21 George Greene <greeneg(a)email.unc.edu> writes: > THERE IS NO SUCH THING AS A FALSE AXIOM. > > That's an axiom. That's a false axiom. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: herbzet on 19 Jul 2010 01:45 Aatu Koskensilta wrote: > herbzet writes: > > > Nah -- implication is truth-preserving, by definition. > > A system may well be consistent even if some of its axioms are false. Sure. > We > are here presupposing the formal language the system is formulated in > has some intended interpretation so that it makes sense to speak of > truth or falsity. Sure, what of it? If this consistent system, so interpreted such that it has false axioms, proves that A -> B, does that mean that A implies B? -- hz
From: Aatu Koskensilta on 19 Jul 2010 01:44
herbzet <herbzet(a)gmail.com> writes: > If this consistent system, so interpreted such that it has false > axioms, proves that A -> B, does that mean that A implies B? No. What of it? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |